Conductance-based Dynamic Causal Modeling: A mathematical review of its application to cross-power spectral densities

TL;DR

This paper provides a comprehensive mathematical review and accessible explanation of conductance-based Dynamic Causal Modeling for cross-spectral densities, including derivations, simulations, and software implementation details.

Contribution

It offers the first detailed, step-by-step mathematical exposition of conductance-based DCM for cross-spectral densities, with practical simulation code.

Findings

Clarifies the mathematical structure of conductance-based DCM

Demonstrates model behavior through simulations

Provides open-source code for practical application

Abstract

Dynamic Causal Modeling (DCM) is a Bayesian framework for inferring on hidden (latent) neuronal states, based on measurements of brain activity. Since its introduction in 2003 for functional magnetic resonance imaging data, DCM has been extended to electrophysiological data, and several variants have been developed. Their biophysically motivated formulations make these models promising candidates for providing a mechanistic understanding of human brain dynamics, both in health and disease. However, due to their complexity and reliance on concepts from several fields, fully understanding the mathematical and conceptual basis behind certain variants of DCM can be challenging. At the same time, a solid theoretical knowledge of the models is crucial to avoid pitfalls in the application of these models and interpretation of their results. In this paper, we focus on one of the most advanced formulations of DCM, i.e. conductance-based DCM for cross-spectral densities, whose components are described across multiple technical papers. The aim of the present article is to provide an accessible exposition of the mathematical background, together with an illustration of the model's behavior. To this end, we include step-by-step derivations of the model equations, point to important aspects in the software implementation of those models, and use simulations to provide an intuitive understanding of the type of responses that can be generated and the role that specific parameters play in the model. Furthermore, all code utilized for our simulations is made publicly available alongside the manuscript to allow readers an easy hands-on experience with conductance-based DCM.